Practice · A straight line is 180 degrees · Sensim Math

Variant 1 medium

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as an angle of $80^\circ$ and another angle measures $50^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a $80^\circ$ angle, angle $b$ , and a $50^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $50^\circ$ angle (on the other side of the line) is angle $a$ .

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1 · Understandwhat's really being asked

Three straight lines cross at one point. Along one line, from left to right, there is a 80-degree angle, angle b, and a 50-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 50-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 80 degrees, b, and 50 degrees add to 180 degrees.
Angle a is across the crossing point from the 50-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b by subtracting the 80 and 50 from the 180-degree straight line. Subproblem 2: angle a is opposite the 50-degree angle across the crossing point, so it equals 50 degrees.

3 · Execute2 carry out the plan

1Find angle b on the straight line

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 80 degrees, b, and 50 degrees make up the 180-degree straight angle. Subtract the two known angles.

b = 180^\circ - 80^\circ - 50^\circ = 50^\circ

The flat line totals 180 degrees; the middle piece is what is left after the other two angles.

2Find angle a as a vertical angle

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 50-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 50 degrees. (You can also see it because a and 50 each complete the same straight line with the same partner angle.)

a = 50^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 50 degrees, angle b = 50 degrees

4 · Reviewdoes it hold up?

Check the line: 80 + 50 + 50 = 180 degrees. And the vertical angle to 50 degrees must also be 50 degrees, so a = 50 degrees is consistent with the crossing.

Another way: Use the same straight-line idea for a (tool 7): along the line carrying the 50-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 80 + 50, so a = 180 - 130 = 50 degrees, matching the vertical-angle result.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡Takeaway. A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!

Variant 2 medium

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as an angle of $100^\circ$ and another angle measures $30^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a $100^\circ$ angle, angle $b$ , and a $30^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $30^\circ$ angle (on the other side of the line) is angle $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three straight lines cross at one point. Along one line, from left to right, there is a 100-degree angle, angle b, and a 30-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 30-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 100 degrees, b, and 30 degrees add to 180 degrees.
Angle a is across the crossing point from the 30-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b by subtracting the 100 and 30 from the 180-degree straight line. Subproblem 2: angle a is opposite the 30-degree angle across the crossing point, so it equals 30 degrees.

3 · Execute2 carry out the plan

1Find angle b on the straight line

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 100 degrees, b, and 30 degrees make up the 180-degree straight angle. Subtract the two known angles.

b = 180^\circ - 100^\circ - 30^\circ = 50^\circ

The flat line totals 180 degrees; the middle piece is what is left after the other two angles.

2Find angle a as a vertical angle

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 30-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 30 degrees. (You can also see it because a and 30 each complete the same straight line with the same partner angle.)

a = 30^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 30 degrees, angle b = 50 degrees

4 · Reviewdoes it hold up?

Check the line: 100 + 50 + 30 = 180 degrees. And the vertical angle to 30 degrees must also be 30 degrees, so a = 30 degrees is consistent with the crossing.

Another way: Use the same straight-line idea for a (tool 7): along the line carrying the 30-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 100 + 50, so a = 180 - 150 = 30 degrees, matching the vertical-angle result.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡Takeaway. A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!

Variant 3 medium

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as a right angle ( $90^\circ$ ) and another angle measures $40^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a right-angle mark ( $90^\circ$ ), angle $b$ , and a $40^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $40^\circ$ angle (on the other side of the line) is angle $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three straight lines cross at one point. Along one line, from left to right, there is a 90-degree angle, angle b, and a 40-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 40-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 90 degrees, b, and 40 degrees add to 180 degrees.
Angle a is across the crossing point from the 40-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b by subtracting the 90 and 40 from the 180-degree straight line. Subproblem 2: angle a is opposite the 40-degree angle across the crossing point, so it equals 40 degrees.

3 · Execute2 carry out the plan

1Find angle b on the straight line

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 90 degrees, b, and 40 degrees make up the 180-degree straight angle. Subtract the two known angles.

b = 180^\circ - 90^\circ - 40^\circ = 50^\circ

The flat line totals 180 degrees; the middle piece is what is left after the other two angles.

2Find angle a as a vertical angle

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 40-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 40 degrees. (You can also see it because a and 40 each complete the same straight line with the same partner angle.)

a = 40^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 40 degrees, angle b = 50 degrees

4 · Reviewdoes it hold up?

Check the line: 90 + 50 + 40 = 180 degrees. And the vertical angle to 40 degrees must also be 40 degrees, so a = 40 degrees is consistent with the crossing.

Another way: Use the same straight-line idea for a (tool 7): along the line carrying the 40-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 90 + 50, so a = 180 - 140 = 40 degrees, matching the vertical-angle result.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡Takeaway. A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!

Variant 4 medium

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as a right angle ( $90^\circ$ ) and another angle measures $25^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a right-angle mark ( $90^\circ$ ), angle $b$ , and a $25^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $25^\circ$ angle (on the other side of the line) is angle $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three straight lines cross at one point. Along one line, from left to right, there is a 90-degree angle, angle b, and a 25-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 25-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 90 degrees, b, and 25 degrees add to 180 degrees.
Angle a is across the crossing point from the 25-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b by subtracting the 90 and 25 from the 180-degree straight line. Subproblem 2: angle a is opposite the 25-degree angle across the crossing point, so it equals 25 degrees.

3 · Execute2 carry out the plan

1Find angle b on the straight line

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 90 degrees, b, and 25 degrees make up the 180-degree straight angle. Subtract the two known angles.

b = 180^\circ - 90^\circ - 25^\circ = 65^\circ

The flat line totals 180 degrees; the middle piece is what is left after the other two angles.

2Find angle a as a vertical angle

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 25-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 25 degrees. (You can also see it because a and 25 each complete the same straight line with the same partner angle.)

a = 25^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 25 degrees, angle b = 65 degrees

4 · Reviewdoes it hold up?

Check the line: 90 + 65 + 25 = 180 degrees. And the vertical angle to 25 degrees must also be 25 degrees, so a = 25 degrees is consistent with the crossing.

Another way: Use the same straight-line idea for a (tool 7): along the line carrying the 25-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 90 + 65, so a = 180 - 155 = 25 degrees, matching the vertical-angle result.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡Takeaway. A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!

Variant 5 medium

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as a right angle ( $90^\circ$ ) and another angle measures $50^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a right-angle mark ( $90^\circ$ ), angle $b$ , and a $50^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $50^\circ$ angle (on the other side of the line) is angle $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three straight lines cross at one point. Along one line, from left to right, there is a 90-degree angle, angle b, and a 50-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 50-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 90 degrees, b, and 50 degrees add to 180 degrees.
Angle a is across the crossing point from the 50-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b by subtracting the 90 and 50 from the 180-degree straight line. Subproblem 2: angle a is opposite the 50-degree angle across the crossing point, so it equals 50 degrees.

3 · Execute2 carry out the plan

1Find angle b on the straight line

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 90 degrees, b, and 50 degrees make up the 180-degree straight angle. Subtract the two known angles.

b = 180^\circ - 90^\circ - 50^\circ = 40^\circ

The flat line totals 180 degrees; the middle piece is what is left after the other two angles.

2Find angle a as a vertical angle

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 50-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 50 degrees. (You can also see it because a and 50 each complete the same straight line with the same partner angle.)

a = 50^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 50 degrees, angle b = 40 degrees

4 · Reviewdoes it hold up?

Check the line: 90 + 40 + 50 = 180 degrees. And the vertical angle to 50 degrees must also be 50 degrees, so a = 50 degrees is consistent with the crossing.

Another way: Use the same straight-line idea for a (tool 7): along the line carrying the 50-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 90 + 40, so a = 180 - 130 = 50 degrees, matching the vertical-angle result.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡Takeaway. A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!

Variant 6 medium

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as an angle of $60^\circ$ and another angle measures $70^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a $60^\circ$ angle, angle $b$ , and a $70^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $70^\circ$ angle (on the other side of the line) is angle $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three straight lines cross at one point. Along one line, from left to right, there is a 60-degree angle, angle b, and a 70-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 70-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 60 degrees, b, and 70 degrees add to 180 degrees.
Angle a is across the crossing point from the 70-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b by subtracting the 60 and 70 from the 180-degree straight line. Subproblem 2: angle a is opposite the 70-degree angle across the crossing point, so it equals 70 degrees.

3 · Execute2 carry out the plan

1Find angle b on the straight line

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 60 degrees, b, and 70 degrees make up the 180-degree straight angle. Subtract the two known angles.

b = 180^\circ - 60^\circ - 70^\circ = 50^\circ

The flat line totals 180 degrees; the middle piece is what is left after the other two angles.

2Find angle a as a vertical angle

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 70-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 70 degrees. (You can also see it because a and 70 each complete the same straight line with the same partner angle.)

a = 70^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 70 degrees, angle b = 50 degrees

4 · Reviewdoes it hold up?

Check the line: 60 + 50 + 70 = 180 degrees. And the vertical angle to 70 degrees must also be 70 degrees, so a = 70 degrees is consistent with the crossing.

Another way: Use the same straight-line idea for a (tool 7): along the line carrying the 70-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 60 + 50, so a = 180 - 110 = 70 degrees, matching the vertical-angle result.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡Takeaway. A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!

Variant 7 medium

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as an angle of $70^\circ$ and another angle measures $60^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a $70^\circ$ angle, angle $b$ , and a $60^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $60^\circ$ angle (on the other side of the line) is angle $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three straight lines cross at one point. Along one line, from left to right, there is a 70-degree angle, angle b, and a 60-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 60-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 70 degrees, b, and 60 degrees add to 180 degrees.
Angle a is across the crossing point from the 60-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b by subtracting the 70 and 60 from the 180-degree straight line. Subproblem 2: angle a is opposite the 60-degree angle across the crossing point, so it equals 60 degrees.

3 · Execute2 carry out the plan

1Find angle b on the straight line

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 70 degrees, b, and 60 degrees make up the 180-degree straight angle. Subtract the two known angles.

b = 180^\circ - 70^\circ - 60^\circ = 50^\circ

The flat line totals 180 degrees; the middle piece is what is left after the other two angles.

2Find angle a as a vertical angle

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 60-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 60 degrees. (You can also see it because a and 60 each complete the same straight line with the same partner angle.)

a = 60^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 60 degrees, angle b = 50 degrees

4 · Reviewdoes it hold up?

Check the line: 70 + 50 + 60 = 180 degrees. And the vertical angle to 60 degrees must also be 60 degrees, so a = 60 degrees is consistent with the crossing.

Another way: Use the same straight-line idea for a (tool 7): along the line carrying the 60-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 70 + 50, so a = 180 - 120 = 60 degrees, matching the vertical-angle result.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡Takeaway. A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!

Variant 8 medium

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as a right angle ( $90^\circ$ ) and another angle measures $60^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a right-angle mark ( $90^\circ$ ), angle $b$ , and a $60^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $60^\circ$ angle (on the other side of the line) is angle $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three straight lines cross at one point. Along one line, from left to right, there is a 90-degree angle, angle b, and a 60-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 60-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 90 degrees, b, and 60 degrees add to 180 degrees.
Angle a is across the crossing point from the 60-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b by subtracting the 90 and 60 from the 180-degree straight line. Subproblem 2: angle a is opposite the 60-degree angle across the crossing point, so it equals 60 degrees.

3 · Execute2 carry out the plan

1Find angle b on the straight line

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 90 degrees, b, and 60 degrees make up the 180-degree straight angle. Subtract the two known angles.

b = 180^\circ - 90^\circ - 60^\circ = 30^\circ

The flat line totals 180 degrees; the middle piece is what is left after the other two angles.

2Find angle a as a vertical angle

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 60-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 60 degrees. (You can also see it because a and 60 each complete the same straight line with the same partner angle.)

a = 60^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 60 degrees, angle b = 30 degrees

4 · Reviewdoes it hold up?

Check the line: 90 + 30 + 60 = 180 degrees. And the vertical angle to 60 degrees must also be 60 degrees, so a = 60 degrees is consistent with the crossing.

Another way: Use the same straight-line idea for a (tool 7): along the line carrying the 60-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 90 + 30, so a = 180 - 120 = 60 degrees, matching the vertical-angle result.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡Takeaway. A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!

Variant 9 medium

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as a right angle ( $90^\circ$ ) and another angle measures $30^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a right-angle mark ( $90^\circ$ ), angle $b$ , and a $30^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $30^\circ$ angle (on the other side of the line) is angle $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three straight lines cross at one point. Along one line, from left to right, there is a 90-degree angle, angle b, and a 30-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 30-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 90 degrees, b, and 30 degrees add to 180 degrees.
Angle a is across the crossing point from the 30-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b by subtracting the 90 and 30 from the 180-degree straight line. Subproblem 2: angle a is opposite the 30-degree angle across the crossing point, so it equals 30 degrees.

3 · Execute2 carry out the plan

1Find angle b on the straight line

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 90 degrees, b, and 30 degrees make up the 180-degree straight angle. Subtract the two known angles.

b = 180^\circ - 90^\circ - 30^\circ = 60^\circ

The flat line totals 180 degrees; the middle piece is what is left after the other two angles.

2Find angle a as a vertical angle

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 30-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 30 degrees. (You can also see it because a and 30 each complete the same straight line with the same partner angle.)

a = 30^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 30 degrees, angle b = 60 degrees

4 · Reviewdoes it hold up?

Check the line: 90 + 60 + 30 = 180 degrees. And the vertical angle to 30 degrees must also be 30 degrees, so a = 30 degrees is consistent with the crossing.

Another way: Use the same straight-line idea for a (tool 7): along the line carrying the 30-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 90 + 60, so a = 180 - 150 = 30 degrees, matching the vertical-angle result.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡Takeaway. A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!

Variant 10 medium

Three straight lines meet at a single point. Along one of the straight lines, one angle is marked as an angle of $110^\circ$ and another angle measures $40^\circ$ . Using the fact that the angles lying along a straight line add up to $180^\circ$ , find the measures of $\angle a$ and $\angle b$ in the figure.

[Figure] Three straight lines cross at one point. Along one straight line through the intersection, from left to right there are a $110^\circ$ angle, angle $b$ , and a $40^\circ$ angle, so these three angles together form the straight angle $180^\circ$ . Directly below the $40^\circ$ angle (on the other side of the line) is angle $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three straight lines cross at one point. Along one line, from left to right, there is a 110-degree angle, angle b, and a 40-degree angle, which together fill the straight angle of 180 degrees. Angle a is directly below the 40-degree angle, on the other side of that line. Find angle a and angle b.

Givens

Three straight lines meet at a single point.
Along one straight line: 110 degrees, b, and 40 degrees add to 180 degrees.
Angle a is across the crossing point from the 40-degree angle (a vertical angle).

Unknowns

Angle b.
Angle a.

Constraints

Angles on a straight line add to 180 degrees.
Angles directly opposite at a crossing (vertical angles) are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b by subtracting the 110 and 40 from the 180-degree straight line. Subproblem 2: angle a is opposite the 40-degree angle across the crossing point, so it equals 40 degrees.

3 · Execute2 carry out the plan

1Find angle b on the straight line

#7 Identify Subproblems 4.MD.C.7

Along that line the three angles 110 degrees, b, and 40 degrees make up the 180-degree straight angle. Subtract the two known angles.

b = 180^\circ - 110^\circ - 40^\circ = 30^\circ

The flat line totals 180 degrees; the middle piece is what is left after the other two angles.

2Find angle a as a vertical angle

#1 Draw a Diagram 4.MD.C.7

Angle a sits directly opposite the 40-degree angle where two straight lines cross. Opposite angles at a crossing are equal, so a is 40 degrees. (You can also see it because a and 40 each complete the same straight line with the same partner angle.)

a = 40^\circ

When two lines cross, the two angles facing each other are mirror-equal in size.

Answer: angle a = 40 degrees, angle b = 30 degrees

4 · Reviewdoes it hold up?

Check the line: 110 + 30 + 40 = 180 degrees. And the vertical angle to 40 degrees must also be 40 degrees, so a = 40 degrees is consistent with the crossing.

Another way: Use the same straight-line idea for a (tool 7): along the line carrying the 40-degree angle, the angle next to a plus a make 180 degrees, and that partner equals 110 + 30, so a = 180 - 140 = 40 degrees, matching the vertical-angle result.

Standardsmin grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting from the 180-degree straight angle to find b and confirming a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the three crossing lines and the opposite (vertical) angle position.

💡Takeaway. A flat line is 180 degrees and angles facing each other across a crossing match - that is all you need to find both a and b!

A straight line is 180 degrees · 10 practice problems

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